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Numberplay: Spooky Action at a Distance

By Pradeep Mutalik June 21, 2010 2:48 pmJune 21, 2010 2:48 pm

Justin Thyme This figure shows the quantum mechanical phenomenon of “Spooky Action at a Distance.” Imagine that these are two pairs of entangled particles that are light years apart. You can choose to “measure” whether the left one is red or blue by putting your cursor on it. At that very moment, the other one turns the opposite color. The “speed” of this interaction is more than 10,000 times the speed of light! Before the measurement is made, the “color” attribute of the particles is undefined.

After my whimsical travels of the last week, it is good to come back to Numberplay (incidentally, we finally have this convenient URL that you can bookmark to get access to all Numberplay posts). In today’s post, I will leave all whimsy aside, and show you that the universe we inhabit is stranger that our fictitious imaginings. Today we will explore a fundamental and almost unbelievable result of Quantum Mechanics (QM) through a set of simple puzzles. This famous result is “Bell’s Interconnectedness Theorem” which was discovered by the physicist John Stewart Bell in 1964. Bell’s theorem has been called by physicist Henry Stapp “the most profound discovery of Science.” It is profound because it shows that the universe has the property of non-locality: two particles can be light years apart, yet they are linked to each other instantaneously. How instantaneously? Recent measurements have shown that this phenomenon occurs more than 10,000 times the speed of light! In order to show that the universe in non-local, Bell showed that a certain simple and obvious inequality – Bell’s inequality – is violated. Amazingly, the only math that you need to understand this profound inequality is contained in the following simple problem.

1. Suppose three friends A, B and C take a test with 100 yes-no questions. If you compare the answers given by A and B, 98 of the 100 are the same. Likewise, if you compare the answers given by B and C, again 98 of the 100 are the same. What is the minimum number of questions that A and C have answered in the same way? If you examine A’s and C’s answers and find that they agree on less than this number, what will you suspect?

Here is an amazing fact: this kind of simple mathematics was missed in the results of Quantum Mechanics for 30 or more years by all the world’s greatest physicists including Einstein and Bohr until it was unearthed by Bell. That’s a truly sobering yet inspiring thought. Who knows, there may be fundamental discoveries that are just waiting for someone to think deeper. Later in this post, I’ll give you a chance to try to make such a discovery yourself!

To understand Bell’s discovery, let us consider the Quantum Mechanical phenomenon of “entanglement.” When two particles are entangled (a situation that can be set up experimentally without much difficulty), and you measure the same property on both, using detectors aligned along the particles’ paths, the results are correlated. If one particle comes out one way, the other comes out either exactly the same (perfectly correlated) or exactly the opposite way (perfectly anti-correlated). In the figure above, we have used the hypothetical “color” property, which is anti-correlated when the detectors are placed in the direction of particles’ motion. The results are absolutely similar when real measurable properties, such as spin, or polarization are used. Quantum Mechanics predicts that even though the two particles are light-years apart, if one comes out blue when you measure its “color,” the other always will always comes out red.

In the famous EPR (Einstein-Podolsky-Rosen) thought experiment, Einstein argued for the presence of “hidden variables.” There are two ways to get this result, he said. The first is that the particles actually communicate faster than light, something he called “spooky action at a distance.” Or else he said there must be “hidden variables” – the particles have made a pre-arranged pact much like the fictitious Obama and Biden did in our Numberplay puzzle a few weeks ago. Effectively, the particles can decide beforehand (i.e. have physical properties that decide) in a random fashion, “you take red, I’ll take blue.” Such an arrangement would explain the result without spooky action at a distance.

And so the matter rested for about 30 years. Then along came Bell and showed that there is, literally, a twist – or a different angle – to this story. In measurements such as spin or polarization, you can orient the detectors for the two particles at different angles, and the results then come out different, but precisely predictable statistically. In our experiment, let us keep one detector unchanged and turn the other through various angles. We find that there is perfect correlation when the detectors are at 0 degrees relative to each other, no correlation (completely random results) when the detectors are at 90 degrees, and perfect anti-correlation when the detectors at at 180 (both detectors in the direction of the particles’ motion, as originally). So if you do the experiment millions of times, the number of times the balls will be that same color is 100 percent for 0 degrees, 50 percent for 90 degrees and no percent for 180 degrees. For every angle in between, there is a precise number that gives the probability of the particles being the same color. This situation can apparently still be handled using a preset arrangement: the particles can carry along with them a table that says for example, “If the detector is at 90 degrees, show up red 50 percent of the time and blue 50 percent of the time.” (Again, note that these could be hidden physical properties and do not presuppose some kind of panpsychic particle intelligence!) If both particles have tables that have entries giving the percentage of red and blue results they have to meet for every direction that the detectors can be set to, then everything will be fine, and there is no need for communication. The observed correlation values can then be met. Or can they? “No,” said Bell. The observed concordance, precisely predicted by QM, is cos2 θ/2 where θ is the angle between the detectors. And what Bell showed, and you can too, is that no system of preset tables can match this.

2. In the figure shown above, assume that the detectors are initially at zero degrees so that both particles are the same color 100 percent of the time. Consider three experiments. In the first one, A, the detector for the left particle is angled slightly in some direction so that the concordance is now 98 percent (look up what angle that will be using the above formula). Now return the left detector to zero, and do experiment B: move the right detector through the same angle in the opposite direction. The concordance should again be the same. Now do experiment C: move both detectors through the above angles at the same time. Note that the A, B and C here are numerically the same as in question 1 above. Based on your answer to question 1, what is the minimum possible concordance? What is the actual observed value, as predicted by the above formula?

Spooky action at a distance is a reality!

3. Bell’s inequality is actually a minimum possible concordance or discordance for every situation. A particularly elegant way to demonstrate Bell’s inequality using just one angle is to do the same three experiments as in question 2 above, but turn the detectors through 120 degrees. What are the minimal and actual values in this instance?

4. Can you construct a function that gives perfect correlation at zero degrees, no correlation at 90 degrees, and perfect anti-correlation at 180 degrees, but does not violate Bell’s inequality?

How do we reconcile spooky action at a distance with Einstein’s Special Theory of Relativity that says that information cannot be transferred at a speed greater than that of light? Well, several physicists have put forth rigorous arguments to show that the correlations discussed above cannot be used to transmit information. So there is no contradiction. Nevertheless the phenomena are truly magical. It seems that, as far as the universe is concerned, the two entangled particles are a “single unified system.” When we make our measurements we decompose this system back into the individual particles, and this internal decomposition takes place faster than the speed of light. Speaking metaphorically, before we make the measurement, the particles have become incorporated, as it were, into a single entity that does not follow the cosmic speed limit. Truly amazing!

5. Now here’s the chance at eternal glory that I promised you. In physics there is a dimensionless number, the reciprocal of the fine structure constant, whose value to the best of our experimental knowledge is 137.035999679 with an uncertainty of 94 in the final two digits. Being dimensionless, this is a pure number with no units. So it is possible that it can be constructed from a combination of small numbers and fundamental numerical constants such as e, π, etc. Can you construct an elegant and not-too-complicated expression that approaches this number? Let’s see who can get closest. Who knows? You may hit the cosmic jackpot!

Interestingly, there are analogues of spooky action at a distance in language as well, which are known as “anaphor resolution” and “preposition stranding.” Consider these sentences:We gave the bananas to the monkeys because they were hungry.
We gave the bananas to the monkeys because they were ripe.
We gave the bananas to the monkeys because they were here.

In the last sentence, does the “they” refer to monkeys or bananas?
And in the following sentence,What did you bring that book that I hate to be read to from out of up through for?
the prepositions are truly entangled! For today’s word challenge, can you come up with humorous examples of linguistic spooky action a distance?

★ Martin Gardner credits.
The last three Numberplay posts honoring Martin Gardner could not have been done without the collaboration of two friends of the great man. You have already met the first of these: Colm Mulcahy (“Card Colm”) who is publishing a new column on the card tricks of Martin Gardner today. Colm went to the trouble of obtaining an I.J.Matrix email address and sending me an email from Dr. Matrix which, as I said, I actually received. Colm posted his fictitious account of the meeting with Dr. Matrix here, and then Dr. I. J. Matrix himself posted a pair of comments last week here and here (these were written jointly by Colm and myself). These postings contain some interesting problems, including a baffling card trick, that readers may want to try.

The other person I am indebted to, is the Irish mathematical enthusiast and author Owen O’Shea who is responsible for the two numerological gems involving the number of days Martin Gardner lived (142 days in 2010, and 34,912 total). Owen seems to be the heir apparent to Dr. I. J. Matrix in numerological acumen. He was among the first, if not the first, to notice that there were 911 days between the 9/11 WTC incident and the 3/11 Madrid attacks. Here is a link to his book “The Magic Numbers of the Professor,” which comes recommended by both Martin Gardner and Clifford Pickover.

June 30, 2010 4:36 a.m. | Updated Well-explained and correct solutions to the questions posed in this post will be highlighted in a couple of weeks and referenced in the subsequent Numberplay column.

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Welcome to our conversation about word games. Here you'll find a new blog post for each day's crossword plus a bonus post for the Variety puzzle. Along with discussion about the day's challenge, you'll get backstage insights about puzzlemaking and occasional notes from The Times's puzzlemaster, Will Shortz.

Deb Amlen is a humorist and puzzle constructor whose work has appeared in The New York Times, The Washington Post, The Los Angeles Times, The Onion and Bust Magazine. Her books, “It's Not P.M.S., It's You” and “Create Your Life Lists” are available where all fine literature is sold.

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About Numberplay, the Puzzle Suite For Math Lovers of All Ages

Numberplay is a puzzle suite that will be presented in Wordplay every Monday. The puzzles, which are inspired by many sources and are reported by Gary Antonick, are generally mathematical or logical problems, with occasional forays into physics and other branches of science. While written for adults, many of the concepts here are suitable for and can be enjoyed by math students of all ages.

Gary Antonick, who has created or edited over 100 logic and math puzzles for The New York Times, secretly believes every math problem can be solved using circles and straight lines. He is a visiting scholar at Stanford University, where he studies mathematical problem solving.

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